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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Stability of the minimal surface system and convexity of area functional
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by Yng-Ing Lee and Mao-Pei Tsui PDF
Trans. Amer. Math. Soc. 366 (2014), 3357-3371 Request permission

Abstract:

We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that $f$ is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at $f$. Then the graph of $f$ is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from $|\bigwedge ^{2} df|\leq \frac {1}{p-1}$ to $|\bigwedge ^{2} df|\leq \frac {1}{\sqrt {p-1}},$ where $p$ is an upper bound of the rank of $df$, and the condition in the 2008 paper of the first author and M.-T. Wang from $\sqrt {det(I+(df)^Tdf)} \leq \frac {43}{40}$ to $\sqrt {det(I+(df)^Tdf)} \leq 2$.
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Additional Information
  • Yng-Ing Lee
  • Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan — and — National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan
  • Email: yilee@math.ntu.edu.tw
  • Mao-Pei Tsui
  • Affiliation: Department of Mathematics and Statistics, University of Toledo, Toledo, Ohio 43606
  • MR Author ID: 278086
  • Email: Mao-Pei.Tsui@Utoledo.edu
  • Received by editor(s): November 23, 2011
  • Published electronically: February 24, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3357-3371
  • MSC (2010): Primary 53A10
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06223-5
  • MathSciNet review: 3192598